1. Direct Fourier Reconstruction
Medical imaging
Group 1
Members:
Chan Chi Shing Antony
Chang Yiu Chuen, Lewis Cheung Wai Tak Steven
Celine Duong
Chan Samson

2. Abstract Shepp-Logan Head Phantom Model Radon Transform
1D Fourier transformed projection slices of different angles
Convert from polar to Cartesian coordinate
Inverse 2D Fourier transform.
Reconstructed image

3. Not that simple!!!
Problem 1: Continuous Fourier Transform is impractical Solution: Discrete Fourier Transform Problem 2: DFT is slow Solution: Fast Fourier Transform Problem 3: FFT runs faster when number of samples is a power of two Solution: Zeropad Problem 4: F1D Radon Function (polar)  Cartesian coordinate but the data now does not have equal spacing, which needs for IF2D Solution: Interpolation

4. Agenda
1.Theory 1.1. Central Slice Theorem (CST) Continuous Time Fourier Transform (CTFT) - > Discrete Time Fourier Transform (DTFT) -> Discrete Fourier Transform (DFT) -> Fast Fourier Transform (FFT) 1.2. Interpolation 2. Experiments 2.1. Basic Number of sensors Number of projection slices Scan angle (<180, >180) 2.2. Advanced Noise Sensor Damage 3. Conclusion 4. References

5. 1. Theory – 1.1. Central Slice Theorem (CST)
Name of reconstruction method:
Direct Fourier Reconstruction
The Fourier Transform of a projection at an angle q is a line in the Fourier transform of the image at the same angle. If (s, q) are sampled sufficiently dense, then from g (s, q) we essentially
know F(u,v) (on the
polar coordinate), and
by inverse transform
can obtain f(x,y)[1].

6. 1. Theory – 1. 1. Central Slice Theorem (CST) – 1. 1
1. Theory – 1.1. Central Slice Theorem (CST) – Continuous Time Fourier Transform (CTFT) - > Discrete Time Fourier Transform (DTFT) -> Discrete Fourier Transform (DFT) -> Fast Fourier Transform (FFT)
CTFT -> DTFT
Description: DTFT is a discrete time sampling
version of CTFT
Reasons: fast and save memory space
DTFT -> DFT
Description: DFT is a discrete frequency
sampling version of DTFT
sampling all frequencies are not
possible
DFT -> FFT
Description: Faster version of FFT
Reasons: even faster

7. 1. Theory – 1. 1. Central Slice Theorem (CST) – 1. 1
1. Theory – 1.1. Central Slice Theorem (CST) – CTFT - > DTFT -> DFT -> FFT
Con’t
DFT -> FFT
Special requirement : Number of samples
should be a power
of two
Solution: Zeropad
How to make zeropad?
In the sinogram, add black
lines evenly on top and
bottom
Physical meaning?
Scan the sample in a bigger
space!

8. 1. Theory – 1.2. Interpolation
Why we need interpolation?
Reasons : Equal spacing for x and y coordinates are required for IF2D
Reasons?
1D Fourier Transform of Radon function is in polar coordinate
Convert to 2D Cartesian coordinate system, x = rcos q and
y = rsin q
Solution: Interpolation

9. 1. Theory – 1.2. Interpolation (con’t)

10. 1. Theory – 1.2. Interpolation (con’t)

11. 2. Experiment – 2.1. Basic – 2.1.1. Number of sensors
The number of sensors decreases
The resolution of the reconstructed images decreases and with low contrast

12. 2. Experiment – 2.1. Basic – 2.1.1. Number of sensors (con’t)

13. 2. Experiment – 2.1. Basic – 2.1.1. Number of sensors (con’t)

14. 2. Experiment – 2.1. Basic – 2.1.1. Number of sensors (con’t)

15. 2. Experiment – 2.1. Basic – 2.1.2. Number of projection slices
As the number of projection slices decreases, the reconstructed images become blurry and have many artifacts
The resolution can be better by using more slices

16. 2. Experiment – 2.1. Basic – 2.1.2. Number of projection slices (con’t)

17. 2. Experiment – 2.1. Basic – 2.1.2. Number of projection slices (con’t)

18. 2. Experiment – 2.1. Basic – 2.1.3. Scan angle (<180, >180)
The image resolution increases as the scanning angle increases
Meanwhile artifacts
reduced

19. 2. Experiment – 2.2. Advanced – 2.2.1. Noise
The noise is added on the sinogram
The more the noise, the more the data being distorted

20. 2. Experiment – 2.2. Advanced – 2.2.2. Sensor Damage
From the sinogram, each s value in the vertical axis corresponds to a sensor
If there is a sensor damaged, then it will appear as a semi-circle artifact

21. 2. Experiment – 2.2. Advanced – 2.2.2. Sensor Damage (con’t)
The more the damage sensors, the lower the quality of the reconstructed images
Could we …
Replace those sensors?
Definitely yes!
Scan the object by 360o instead of 180o?
No.

22. 3. Conclusion
Shepp-Logan Head Phantom Model
Radon Transform
1D Fourier transformed projection slices of different angles
Convert from polar to Cartesian coordinate
Inverse 2D Fourier transform.
Reconstructed image
Direct Fourier Reconstruction uses short computation time to give a good quality image, with all details in the Phantom can be conserved
The resolution is high and even there is little artifact, it is still acceptable.
To make the reconstructed images better, we can
use more sensors
use more projection slices
scan the Phantom more than 180o
add filters to eliminate noise
Replaced all damaged sensors.

23. 4. Reference
1. Yao Wang, 2007, Computed Tomography, Polytechnic University
2. Forrest Sheng Bao, 2008, FT, STFT, DTFT, DFT and FFT, revisited, Forrest Sheng Bao,

24. Thank you